Coaxial cable ($Z_0=(2\pi {)}^{-1}\sqrt{\mu /\epsilon }\log (b/a)$)

A coaxial cable has an inner conductor surrounded by a concentric outer conductor, with dielectric between them. In the ideal homogeneous lossless Transverse electromagnetic mode, electric field is radial and magnetic field circles the axis; the return current flows on the outer conductor. For inner radius $a$ and inner shield radius $b$, its characteristic impedance isFor air dielectric this is approximately $60\log (b/a)$ ohms. A cable's characteristic impedance describes travelling waves, not its DC wire resistance. Finite conductivity, imperfect shielding and higher modes limit the ideal description.

For a coaxial cable with inner and outer radii $a,b$, a radial interface $c$, and permittivities $ϵ,{ϵ}^{\prime }$, define $T=\ln (c/a)/ϵ+\ln (b/c)/{ϵ}^{\prime }$. With terminal electric potential difference $V$, the electric displacement field is $D_{\rho }=V/(T\rho )$. The electric field is $E_{\rho }=V/(T{ϵ}_i\rho )$ in each layer $i$. If the outward and return electric currents have magnitudes $I$, the magnetic field strength in the gap is $H_{\phi }=I/(2\pi \rho )$ and the magnetic field is $B_{\phi }={\mu }_{i}I/(2\pi \rho )$.

The proof uses Gauss's law without free interface electric charge, the electric potential difference integral, and the magnetostatic Ampère law. The axial Poynting vector is $VI/(2\pi T{ϵ}_i{\rho }^2)$, so its cross-sectional surface integral is $VI$, independently of the radial material layering and magnetic permeabilities. The empty inner and exterior regions have zero electric field and magnetic field for the regular solution with bounded exterior electric potential and no applied magnetic field.

A small longitudinal electric field drives conductor current in a lossy coaxial line. Its cross product with the azimuthal magnetic field sends energy radially into both conductors, where it becomes heat; forward power decreases with distance.

For ideal coaxial conductors of radii $a<b$ filled with a homogeneous permittivity $ϵ$ and magnetic permeability $\mu$, Gauss's law gives $E_r=\lambda /(2\pi ϵr)$ and hence $C^{\prime }=2\pi ϵ/\ln (b/a)$. The magnetic field $H_{\phi }=I/(2\pi r)$ has energy per unit length $L^{\prime }{I}^2/2$, giving $L^{\prime }=\mu \ln (b/a)/(2\pi )$. These are the external-field parameters when internal magnetic penetration into the conductors is neglected. Their product is $L^{\prime }{C}^{\prime }=\mu ϵ$, and the characteristic impedance is $\sqrt{L^{\prime }/C^{\prime }}=\sqrt{\mu /ϵ}\ln (b/a)/(2\pi )$.